It's an ablative absolute. (Thus, "cognitō" is actually neuter, since "cōnsilium" is.) "But when their plan became known [lit. "with their plan known"], he gave the money and other things to the sailors..."

Shouldn't 'Nostrum' be 'Nostra' - to agree with 'Vita' seeing as: vita nostra servata est. 'Nostrum' is an adjective so shouldn't it agree with 'vita'. Also the sentence is using a passive voice construction.

The reason why I say 'nostrum' - which is an alternative to 'nostri' (also genitive) is that, as a partitive genetive it takes that alternative form (?) - and I'm observing that the alternative genitive 'nostrum' has the same ending as 3d declension genitive plural nouns / adjectives.

I'm probably being very deliberative and simple minded...!

My original question was in what circumstances do you use 'nostrum' as opposed to 'nostri'.

The genitives of the personal pronouns nostri and vestri are objective, e.g., cupidus nostri ("desirous of us," "wanting us"); the genitives of the personal pronouns nostrum and vestrum are partitive, e.g., unus nostrum ("one of us").

I read "vita ominium nostrum" literally as "the life of all of us," with both omnium and nostrum functioning pronominally, not adjectivally.

As you said, both nostrum and nostri are the genitive plural forms of the personal pronoun nos (we). Nostrum is a partitive genitive: unus nostrum (one of us). Nostri is an objective genitive: cupidus nostri (desirous of us, wanting us).

The phrase "omnes nostrum" could be translated as "all of us", e.g., omnes nostrum adsumus (all of us are here). Omnibus nostrum could be translated as "to all of us". e.g., pecuniam omnibus nostrum dedit (he gave money to all of us). Omnium nostrum could be translated as "of all of us", e.g., vita omnium nostrum servata est (the life of all of us has been saved).